Precise Rates in the Generalized Law of the Iterated Logarithm in ${\mathbb{R}}^m$ 
Received:February 06, 2017 Revised:August 04, 2017 
Key Word:
precise rates law of iterated logarithm complete convergence i.i.d. random vectors

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.61662037) and the Scientific Program of Department of Education of Jiangxi Province (Grant Nos.GJJ150894; GJJ150905). 

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Abstract: 
Let \{$X$, $X_n$, $n\ge 1$\} be a sequence of i.i.d. random vectors with ${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$ and ${\rm Cov}(X,X)=\sigma^2I_m$, and set $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. For every $d>0$ and $a_n=o((\log\log n)^{d})$, the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of ${\mathbb{P}}(S_n\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$. 
Citation: 
DOI:10.3770/j.issn:20952651.2018.01.010 
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